Find f.

f ' (x)= 4/{square root(1-2[squared])}, f(1/2)= 1

Any insight as to how to handle this would be wonderful :)

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- May 19th 2008, 12:47 AMCALsunshineHelp with Anti-derivative problem
Find f.

f ' (x)= 4/{square root(1-2[squared])}, f(1/2)= 1

Any insight as to how to handle this would be wonderful :) - May 19th 2008, 02:00 AMmr fantastic
- May 19th 2008, 03:09 AMSoroban
Hello, CALsunshine!

I assume there is a typo . . .

Quote:

$\displaystyle f'(x)\:=\:\frac{4}{\sqrt{1-x^2}},\quad f\left(\frac{1}{2}\right)\:=\: 1$

Find $\displaystyle f(x).$

Integrate: . $\displaystyle f(x) \;=\;\int\frac{4}{\sqrt{1-x^2}}\,dx \quad\Rightarrow\quad f(x) \;=\;4\sin^{\text{-}1}(x) + C$

Since $\displaystyle f\left(\frac{1}{2}\right) \,=\,1$, we have: .$\displaystyle 1 \;=\;4\sin^{\text{-}1}\left(\frac{1}{2}\right) + C$

. . Then: .$\displaystyle 1 \;=\;4\left(\frac{\pi}{6}\right) + C \quad\Rightarrow\quad C \:=\:1 - \frac{2\pi}{3}$

Therefore: .$\displaystyle f(x)\;=\;4\sin^{\text{-}1}(x) + 1 - \frac{2\pi}{3}$